Use the Factoring Polynomials Calculator to quickly find the factored form of any quadratic polynomial in the form $ax^2 + bx + c$. This tool utilizes the quadratic formula to determine the roots, providing an immediate, accurate, and step-by-step factorization.
Factoring Polynomials Calculator
Enter the coefficients for the polynomial $ax^2 + bx + c$.
Factored Form:
Factoring Polynomials Formula
For a general quadratic polynomial $P(x) = ax^2 + bx + c$, the factorization over real numbers is determined by finding its roots ($r_1$ and $r_2$) using the quadratic formula:
Factored Form: $P(x) = a(x – r_1)(x – r_2)$
Formula Source: Wikipedia: Quadratic Formula
Variables
The Factoring Polynomials Calculator requires three inputs:
- Coefficient ‘a’ ($x^2$): The number multiplying the $x^2$ term. If this is 1, you can leave the term implicit (e.g., $x^2 + 5x + 6$). Must not be zero for a quadratic.
- Coefficient ‘b’ ($x$): The number multiplying the linear $x$ term.
- Constant ‘c’: The independent term, or the value of the function when $x=0$.
What is Factoring Polynomials?
Factoring a polynomial is the process of breaking down a polynomial expression into a product of simpler polynomials, typically linear or irreducible quadratic factors. It is essentially the reverse process of multiplication (or expanding) and is a fundamental concept in algebra.
For quadratic polynomials (degree 2), factoring allows us to easily find the roots (or zeros) of the function—the values of $x$ for which the polynomial equals zero. If a polynomial can be factored into linear terms, $a(x-r_1)(x-r_2)$, then the roots are simply $r_1$ and $r_2$.
This calculator focuses on factoring quadratic polynomials using the quadratic formula, which works reliably for all cases (including those with irrational or complex roots), making it more powerful than factoring by inspection or grouping alone.
How to Calculate Factoring Polynomials (Example)
Let’s factor the polynomial $x^2 + 7x + 12$.
- Identify Coefficients: In $x^2 + 7x + 12$, we have $a=1$, $b=7$, and $c=12$.
- Calculate the Discriminant ($\Delta$): The discriminant is $b^2 – 4ac$. $$\Delta = 7^2 – 4(1)(12) = 49 – 48 = 1$$
- Find the Roots ($r_{1,2}$): Since $\Delta > 0$, there are two real roots. $$r_{1} = \frac{-7 + \sqrt{1}}{2(1)} = \frac{-7 + 1}{2} = -3$$ $$r_{2} = \frac{-7 – \sqrt{1}}{2(1)} = \frac{-7 – 1}{2} = -4$$
- Determine the Factored Form: The form is $a(x-r_1)(x-r_2)$. $$1 \cdot (x – (-3))(x – (-4)) = (x+3)(x+4)$$ The factored form is $\mathbf{(x+3)(x+4)}$.
Frequently Asked Questions (FAQ)
- What does the discriminant ($b^2 – 4ac$) tell me about the factors?
If the discriminant is positive ($\Delta > 0$), there are two distinct real factors. If it is zero ($\Delta = 0$), there is one repeated real factor. If it is negative ($\Delta < 0$), there are no real factors, only complex conjugate factors. - Can this calculator factor polynomials with a degree higher than 2?
No, this calculator is specifically designed for quadratic polynomials ($ax^2 + bx + c$). Factoring higher-degree polynomials often requires more complex techniques like the Rational Root Theorem or synthetic division. - What should I input if the polynomial is $3x^2 – 9$?
For $3x^2 – 9$, the missing $x$ term means $b=0$. You should input $a=3$, $b=0$, and $c=-9$. - Why is finding the roots the same as factoring?
The Factor Theorem states that $(x-r)$ is a factor of a polynomial $P(x)$ if and only if $r$ is a root of $P(x)$, meaning $P(r)=0$. Finding the roots directly gives you the linear factors.